The Gross-Pitaevskii equations of a static and spherically symmetric condensate of gravitons

TL;DR

This paper models a graviton condensate as a Bose-Einstein condensate using Gross-Pitaevskii equations, revealing that such a condensate forms a gravastar with a sub-Planckian cosmological constant, resembling a quantum, horizonless black hole alternative.

Contribution

It derives two Gross-Pitaevskii equations for a graviton condensate, showing the effective geometry is a gravastar with specific quantum properties.

Findings

The condensate forms a gravastar with a DeSitter interior.

The effective geometry has no horizon but matches the Schwarzschild radius.

The condensate is always quantum and weakly coupled for super-Planckian masses.

Abstract

In this paper we consider the Dvali and G\'omez assumption that the end state of a gravitational collapse is a Bose-Einstein condensate of gravitons. We then construct the two Gross-Pitaevskii equations for a static and spherically symmetric configuration of the condensate. These two equations correspond to the constrained minimisation of the gravitational Hamiltonian with respect to the redshift and the Newtonian potential, per given number of gravitons. We find that the effective geometry of the condensate is the one of a gravastar (a DeSitter star) with a sub-Planckian cosmological constant, for masses larger than the Planck scale. Thus, a condensate corresponding to a semiclassical black hole, is always quantum and weakly coupled. Finally, we obtain that the boundary of our gravastar, although it is not the location of a horizon, corresponds to the Schwarzschild radius.